How do I turn a scene into a 3-color...logo or cartoon looking image?
Oh...you want to do that to a scene? Whoops. Well this is how it would work. You would still use nodes, but instead of using the Material nodes, use the Compositing Nodes (the tab with the two pictures at the bottom of the Node Editor.)

This is what the scene would normally look like.


Here's a few different effects you could try out.

A Persona-esque effect:


A Noir (or Gameboy) effect:


Or just a totally simplified color scheme:


Again:
Make sure your film is transparent. Open the tab with a photo camera on it in Properties, scroll down to "Film," and check the transparency box.

Feel free to mess around with the ColorRamps until you get a good result. Any change makes a big difference! Just remember to keep it on "Constant."